It depends on how competitive you want to be (gotta win, or just have fun), at what level (local, divisional, or national), with what organization (SCCA, NASA), and at what budget (got a family to support on one income versus you just won the Powerball). I do autocross mine and have modded it heavily for that purpose, and locally am very competitive. (I just set FTD last Sunday though that was a fluke of the weather. I'm always in the hunt for a class win and I've finished second for the season overall in a very competitive Street Mod class 3 yrs in a row. But when the big boys who run Nationally come in from out of town, I can't really touch them.) So I think I can speak on this.
First, you will not be competitive for some time regardless of what you drive until you learn to autocross. The number one piece of speed equipment is the driver. Unless you have Michael Schumacher in your genes, it will take at least a couple years to learn how to read the course, find the right line, not go too fast in the slow parts and too slow in the fast parts, learn car set up, and so on. Which is a good thing because it will take that long to make the mods you need.
Stock, car is not competitive unless you run against hacks and beaters. Sorry but it's true. It weighs too much, understeers relentlessly, and will be classed against more nimble vehicles. But you may as well run stock while you learn to drive, unless you have been doing this already in other cars.
You will want to get into a class that allows some level of preparation in order to unlock some of the car's potential, including modified boost control and bolt-on suspension. Candidates include Street Prepared (used to not allow boost control but I think that has changed), and Street Touring Unlimited (be prepared to run against Evos and STIs). The ultimate for the VR4 is Street Modified. Anything goes so long as the engine is badged the same as original, the suspension attaches to the original holes, metal panels that contain glass are still metal, you do not cut or weld the frame or firewall, and the full interior is retained. The car can be competitive at this level locally if you are willing to spend a fair amount of dough. If you have national aspirations, leave the VR4 at home and pick a different car that weighs about 800 lb less because that is its principal disadvantage.
Get a copy of the rule book so you know what you can modify for a given class. One wrong mod can bump you up so that you are bringing a duck to a cock fight. In general you will need wider wheels, race tires (Hoosiers or Kumhos) unless you run a street tire class (note stock classes are not limited to street tires in SCCA), much stiffer springs, adjustable shocks, lots more negative camber up front (camber plates or bolts), poly bushings, much heavier rear swaybar, and an alignment. Disconnecting and removing the rear steer is worthwhile. You will want to move the battery to the trunk and get a CF hood to take weight off the front. A heavy duty clutch for launches is a must, as the stock clutch will be toast after one season. A rear limited slip diff is a huge asset. Start with Eclipse LSDs and after you wear out a couple of those you'll bite the bullet and get a mechanical unit (Kaaz). Do whatever you want to the motor as allowed per class. Bigger intercooler and exhaust for sure, electronic boost controller, the usual. Stay away from big turbos that build boost at 4000 RPM. What you need is torque to dig out of corners and good mid-range throttle response for the many transitions, so smaller turbos are better. Or ball bearing turbos if you have the dough. Race seat and harness will be helpful if you are generating high cornering forces, but in a pinch a baby seat clip on the lap belt and a utility strap across the upper chest can do a decent job of holding you in place while you fling it.
There is just a whole lot more if you want to keep pursuing it. Just realize that driver talent is a huge part of the success equation, and at the end of the day the VR4 is not necessarily an optimal choice. But it can be a lot of fun and a real attention-getter especially if you get it to go faster than anyone expects. And have fun. Nothing ruins a good time racing as taking it too seriously.
Mike R.